How Far Is Far?

by Lorna Simmons
Printed in Reflections: October, 1999.

Have you ever wondered how distances in the universe are calculated?
Have you ever been fascinated by the methods used by astrophysicists and
cosmologists to search the great unknown?

If we are going to take such an exciting journey, it always starts with
that first step. Although today, there are laser measurements and
radar calculations in use for distances in the Solar System. These
measurements would hardly work for the farther objects in the universe,
which often are the distances cosmologists really want to know more about.

With the very first rungs of the cosmological distance ladder, after radar
and lasers, of course, everything heats up. Tempers flare.
Friendships are lost. Personal preferences of the various cosmologists
roil the waters of controversy. Skirmishes turn into battles into
wars. So much in cosmology depends upon the measurements of these
distances that winning becomes all in the minds of the combatants.
There is ever present the incredibly difficult task to determine absolutely
correct uncertainties for each of the observations. The errors are
usually underestimated, seldom overestimated. Perhaps I exaggerate
too much, but sometimes it seems like a battle. Cosmology is a hot
topic. Reputations are at stake.

At this moment, we are on the verge of a breakthrough where precise observations
will be able to reveal the secrets of distances in the universe.
Because the distance scale leads to calculations of the age of the Universe,
its rate of expansion, its density, and inevitably its destiny, such measurements
are of immense importance in cosmology. However, that task is extremely
arduous. On the ladder, every rung depends on previous lower rungs.
A tiny error in the calibration of each rung will magnify the computational
errors of measurements farther out on the ladder. Pretty messy.

It is best to split up the distance indicators into groups - primary and
secondary. The indicators of the primary groups are calibrated from
measurements in our very own Milky Way, and the secondary indicators rely
on at least one of these primary indicators. Therefore, the secondary
indicators become increasingly less reliable as we get farther out on the
distance scale.

Primary Group

To begin our journey up the cosmological distance ladder, the parallax
method of determining distance is fundamental to all of the others.
It is the most certain of methods for determining distances outside of
the Solar System. The parallax method uses the elliptical orbit of
the Earth around the Sun to determine the baseline for its measurements
of the slight changes in the placement of the nearest stars. Distances
to stars within approximately 40 parsecs (pc) can easily be calculated
using a simple trigonometry. An angle is measured in arcseconds at
one point of Earth’s orbit, and six months later, at the opposite position
of Earth’s orbit, another measurement of the angle in arcseconds is taken.
The slight difference of the observed movement of the celestial object
with relation to the seemingly fixed stars creates an angle which subsequently
provides the distance to that object in parsecs, one parsec equaling 3.2616
light years (ly). Note that the recent Hipparcos satellite observations
seem to be increasing the range of such parallax measurements to at least
100 parsecs.

Using parallax methods, we next move to the standard candles, of which
the Cepheid variables are of first importance. Cepheid variables
are large and very luminous yellow giant stars or supergiants with regular
periods of pulsation from about 1 to nearly 70 days. Over 700 Cepheids
have been detected in our own Milky Way and a few thousand Cepheids can
be seen in our Local Group of galaxies. Changes in brightness of
Cepheid variables are accompanied by changes in their stellar temperatures
and in their radii. A ratio exists between their periods of pulsation
and their light variation (period-luminosity). The distance to the
Cepheids is calculated by measuring their brightness and their periods
of brightening. In other galaxies, noting the periods of the brightening
would help determine the distances to such galaxies. Moderately great
distances in the universe can be calculated using Cepheids, although the
uncertainties are still very large.

RR Lyrae stars, which are also pulsating variable stars similar to the
Cepheid variables, are old population II stars and are common in globular
clusters. Their metallicity and absolute magnitudes are compared,
making RR Lyrae stars a major rung up in the cosmological distance ladder.
However, because RR Lyrae have low luminosities they cannot be used much
farther than M31, limiting their use.

Many other similar stellar methods exist. Statistical parallaxes
measure radial velocities and the proper
motions of groups of stars to calculate distances in the universe to approximately
500 pc. Other similarly luminous stars, such as W Virginis stars
and Mira variables also have a period-luminosity relation. The brightest
red giants in globular clusters are helpful in measuring the distance ladder
as are the eclipsing binaries where both spectroscopy and photometry are
useful in recording the colors. Farther out on the ladder, cosmologists
obtain exceedingly accurate measurements of the nearest open clusters using
the Cepheid Variables in these clusters. And the Cepheids in these
clusters will open the possibility for a new and better calibration of
the important period-luminosity relation for Cepheids.

Another way of measuring the universe is the converging-point method, using
the proper motion of open clusters. The average velocities of open
clusters is compared to that of the Sun and is large with respect to the
proper motions of stars. Hence, those stars can be thought of as
moving toward the same point in the universe. For example, the distance
to the Hyades can be determined using this method. Next, another
method called “main-sequence fitting” compares the apparent magnitude
with the absolute magnitude of a star, permitting the distance to that
cluster to be calculated.

Secondary Group

There are a number of secondary distance indicators for measuring even
farther out into the universe. First, there is the Tully-Fisher relation
which uses spiral galaxies to calculate distances easily; well, almost
easily for professional cosmologists. There are several versions
of the Tully-Fisher relation. The first type measures the absolute
magnitude of the galaxy, while the second type uses the infrared spectrum.
The third type determines the rotational velocity of the galaxy.
Another method, called the Fundamental Plane, finds similar relations for
elliptical galaxies.

Supernovae, particularly Type Ia, are frequently used and are exceedingly
promising standard candles. Supernovae are some of the brightest
objects in the Universe and are visible to exceedingly high redshifts.
The distances can be estimated because Type Ia supernovae have extraordinarily
precise variations in light and color variations. The distribution
of absolute magnitudes at maximum light seems to be very narrow.
A few cases where Type Ia supernovae have been unusually red or spectroscopically
peculiar lack this relation. Most recently, Type Ia supernovae have
shown that the universe is
accelerating in its expansion. So far, the Type Ia supernovae have
given promising results in this study. Time will tell.

Next, the Sunyaev-Zel’dovic effect appears when the Cosmic Microwave Background
Radiation (CMBR) passes through galactic clusters which contain extremely
hot intracluster gas. The accompanying dimming of the CMBR is measured
as resulting from the scattering of the CMBR photons in the intracluster
gas and results in a measure of the physical depth of the cluster.
This, combined with the angular extension of the cluster, gives its distance.
This method reaches far into the universe.

Finally, there are Quasi-stellar Objects (QSOs) or Quasars which have undergone
gravitational lensing. The time delay is measured between the fluctuations
in the multiple images of these distant objects. This probes into
the very distant universe and perhaps will become useful in future cosmological
tests. Using QSOs is a problem because this method is dependent on
the measuring of the lensing mass in front of the QSO which leads to quite
large uncertainties in the resulting calculations.

In addition to all of the above, there are many other secondary methods
for determining cosmic distances. However, there is much greater
scatter in their measurements making the accuracy much less certain.

In closing, it is important to mention the doppler effect shown by blueshift
and redshift calculations which are part of the cosmologists’ tools to
determine velocities of objects, both toward and away from the observer,
within the Milky Way. Outside of the Milky Way, redshifts and blueshifts
are calculated within our Local Group of galaxies to determine relative
radial velocities toward and away from our galaxy. Extragalactic
sources, including quasars, also show a doppler effect which results from
the expansion of the universe. Recessional velocities of distant
galaxies can be exceedingly large and the expression for redshift of “z”
is used in these cases. Redshifts, themselves, exhibit a photon energy
loss in radiation for overcoming the effects of recession or expansion.
Using Hubble’s law for distance determination, the measurements of galactic
redshifts are also used to calculate the distances of galaxies. Redshifts
can also be produced by the presence of strong gravitational fields, as
predicted by
Einstein’s General Theory of Relativity. It should be understood
that, although galactic redshifts can
be interpreted as caused by the relativistic doppler effect alone, both
the expansion and the gravitation fields of the universe are involved.
Therefore, we should be aware that redshifts may, alternatively or in combination,
be considered as doppler redshifts, gravitational redshifts, and/or cosmological
redshifts.

So next time you wonder about the accuracy of the methods for determining
cosmic distances, rest assured that the cosmologists are handling it just
fine - if we can ever get them away from the battlefield.